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A Mixture Gamma Distribution to Modelthe SNR of Wireless Channels

Saman Atapattu, Student Member, IEEE, Chintha Tellambura, Fellow, IEEE, and Hai Jiang, Member, IEEE

Abstract— Composite fading (i.e., multipath fading and shad-owing together) has increasingly been analyzed by means of theK channel and related models. Nevertheless, these models dohave computational and analytical difficulties. Motivated by thiscontext, we propose a mixture gamma (MG) distribution forthe signal-to-noise ratio (SNR) of wireless channels. Not only isit a more accurate model for composite fading, but is also aversatile approximation for any fading SNR. As this distributionconsists of N (≥ 1) component gamma distributions, we showhow its parameters can be determined by using probabilitydensity function (PDF) or moment generating function (MGF)matching. We demonstrate the accuracy of the MG model bycomputing the mean square error (MSE) or the Kullback-Leibler(KL) divergence or by comparing the moments. With this model,performance metrics such as the average channel capacity, theoutage probability, the symbol error rate (SER), and the detectioncapability of an energy detector are readily derived.

Index Terms— Fading channels, mixture of gamma distribu-tions, signal-to-noise ratio (SNR).

I. INTRODUCTION

Fundamental wireless propagation effects include macro-scopic (large scale or slow) fading and microscopic (smallscale or fast) fading. Macroscopic fading results from the shad-owing effect by buildings, foliage and other objects. Micro-scopic fading results from multipath, which occurs in indoorenvironments, and also both macrocellular and microcellularoutdoor environments [2]. Shadowing can significantly impactsatellite channels, point-to-point long distance microwave linksand macrocellular outdoor environments [3]. Both microscopicand macroscopic fading together are modeled by compositeshadowing/fading distributions, of which Rayleigh-lognormal(RL) and Nakagami-lognormal (NL) are the two most commonmodels [2]. But the probability density function (PDF) of thesetwo composite models are not in closed form, making perfor-mance analysis of some applications difficult or intractable.Hence, several other composite models have been developedincluding the Suzuki distribution, the K and generalized-K(KG) distributions, the G- distribution, and the Gamma distri-bution [4]–[8]. Note that these models are approximations ofthe RL and NL models.

Manuscript received November 25, 2010; revised June 1, 2011 and August11, 2011; accepted August 16, 2011. The editor coordinating the review of thispaper and approving it for publication was Dr. Man-On Pun. This work wassupported by the Izaak Walton Killam Memorial Scholarship at the Universityof Alberta, Canada, the Alberta Innovates - Technology Futures, Alberta,Canada, and the Natural Science and Engineering Research Council (NSERC)of Canada. The authors are with the Department of Electrical and ComputerEngineering, University of Alberta, Edmonton, AB, Canada T6G 2V4 (e-mail:atapattu@ualberta.ca, chintha@ece.ualberta.ca, hai1@ualberta.ca).

A. Performance Analysis Based on K and KG Models

The use of K and KG models for performance analysis hasgreatly increased recently (see [9]–[19], among many others).For instance, the statistics of signal-to-noise ratio (SNR), theaverage channel capacity and the bit error rate (BER) areanalyzed in [9], [10]. The outage performance, the averageBER, and the channel capacity of several adaptive schemes arederived in [11]. The average output SNR, amount of fading andoutage probability of different diversity receivers are derivedin [12]. The closed-form BER is derived for (post-detection)equal gain combining (EGC) in [13]. The performance of dual-hop non-regenerative relays and multihop regenerative relaysis analyzed in [14]–[16]. The average BER of orthogonalfrequency-division multiplexing (OFDM) systems is evaluatedin [17]. The ergodic capacity of multiple-input multiple-output (MIMO) systems is investigated in [18]. In [19], theperformance of an energy detector is analyzed. These studiesand others show the importance of K and KG models.

B. Limitations of K and KG Models

While RL and NL do not have closed-form PDFs, the K,KG and G- models do. Nevertheless, their PDFs include spe-cial functions (e.g., modified Bessel functions). Consequently,mathematical complications arise in the evaluation of wirelessperformance metrics. For instance, the cumulative distributionfunction (CDF) of the KG model is derived in [9] by usinggeneralized hypergeometric functions. The computation ofsuch functions can be difficult as their series expressions maygive rise to numerical issues. Asymptotic expansions mayhence be required for certain ranges of the parameters andthe variables. Moreover, the PDF of a sum of SNRs (requiredin maximal ratio combining [MRC]) is intractable. As well,even numerical methods for MRC by using the characteristicfunction approach is quite difficult due to the Whittakerfunction [8]. To avoid these difficulties, KG random variables(RVs) are approximated by Gamma RVs in [8], and the PDFof the sum of independent KG RVs is further approximatedby PDF of another KG RV [20]. These approximations arebased on moment matching.

C. Mixture Gamma Distribution

While performance evaluation over composite channels ishighly important, the use of K, KG and G models is notwithout analytical and/or computational difficulties. Hence, wedevelop an alternative approach by using the mixture gamma(MG) distribution [21], [22].

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This distribution avoids the problems mentioned above dueto several reasons. First, since it is a linearly weighted sumof gamma distributions, it inherits several advantages of thegamma distribution. For example, the moment generatingfunction (MGF) and CDF, which are required in wirelesssystem analysis, have mathematically tractable expressions.Second, this distribution can approximate not only compositefading channels, but also any small-scale fading channels.Third, high accuracy is possible by adjusting the parameters.Overall, by using the MG model, performance of any wirelesssystems over a variety of fading channels can be analyzed ina unified framework.

D. Contributions

Our main contributions are summarized as follows.

• We propose an MG distribution to model the SNR ofwireless channels. Although primarily intended to modelcomposite shadowing/fading channels, this distribution isnevertheless effective for many other existing small-scalefading channels as well. The MG distribution is discussedin the statistics literature [21], [22]. The use of such adistribution to model wireless channels is new as far aswe know.

• In the existing literature [6]–[8], KG, G and Gammadistributions are used to approximate the compositeshadowing/fading models. The MG distribution is moreaccurate than all those. The accuracy is measured bythe mean square error (MSE), the Kullback-Leibler (KL)divergence, or by a comparison of the moments or thePDFs.

• The PDFs of K, KG, η-µ, Nakagami-q (Hoyt), κ-µ, or Nakagami-n (Rician) distributions contain specialfunctions, and thus, performance analysis is complicatedor intractable. The MG model, a linear combination ofgamma distributions, offers a solution. Thus, the distri-butions mentioned above can be approximated by the MGmodel, facilitating performance analysis.

• Another advantage of the MG distribution is the sim-plicity of performance analysis. Specifically, its CDF,MGF and moments are readily mathematically tractable.Typical performance analysis scenarios are derived undera unified framework. Therefore, case-by-case analysisof different channel models is unnecessary. Moreover,performance metrics, such as the average capacity, theoutage probability, the symbol error rate (SER), andparticularly, performance of an energy detector (essentialfor future cognitive radio networks), are derived by usingthe unified framework.

The rest of the paper is organized as follows. The MG dis-tribution is described in Section II. Several common wirelesschannels are represented by using the MG model in SectionIII. In Section IV, the accuracy of the MG representation ofcomposite fading channels and small-scale fading channelsis examined. Performance analysis and the numerical resultsfrom the unified framework are shown in Section V. Theconcluding remarks are in Section VI.

II. THE MG WIRELESS CHANNEL MODEL

We start with the SNR distribution, which is required foranalysis of wireless communication systems. The instanta-neous received SNR and the average SNR are denoted by γand γ, respectively.

A. Probability Density Function (PDF)

In [23], it is shown that any function f(x), where x ∈(0,∞) and limx→+∞ f(x) → 0, can be given as f(x) =

limu→+∞ Su(x) where Su(x) := e−ux∑∞

k=0(ux)k

k! f(ku

),

u > 0. Thus, an arbitrarily close approximation to f(x) canbe obtained by increasing the number of terms in the mixture[24]. Note that Su(x) is a weighted sum of gamma PDFs. Thisresult provides the motivation for using the MGF distributionto represent any wireless SNR models.

Therefore, we propose to use the following MG distributionto approximate the PDF of γ as

fγ(x) =N∑i=1

wifi(x) =N∑i=1

αixβi−1e−ζix, x ≥ 0 (1)

where fi(x) =ζβii xβi−1e−ζix

Γ(βi)is a standard Gamma distri-

bution, Γ(·) is the gamma function, wi = αiΓ(βi)

ζβii

, N isthe number of terms, and αi, βi and ζi are the parametersof the ith Gamma component. Further,

∑Ni=1 wi = 1 as∫∞

0fγ(x)dx = 1. The special case N = 1 reverts to Rayleigh

and Nakagami-m fading. Discussion of how to choose N isprovided in Section IV. Note that formula (1) can approximatethe PDF f(x) of any positive random variable.

B. Cumulative Distribution Function (CDF)

The CDF of the MG distribution can be evaluated asFγ(x) =

∫ x

0fγ(t)dt to yield

Fγ(x) =N∑i=1

αiζ−βi

i γ (βi, ζix) (2)

where γ(·, ·) is the lower incomplete gamma function definedas γ(a, ρ) ,

∫ ρ

0ta−1e−tdt [25, eq. (8.350.1)]. This function

is readily available in mathematical software packages.

C. Moment Generating Function (MGF)

The MGF of MG distribution, Mγ(s), can be evaluatedas Mγ(s) = E(e−sx) where E(·) denotes the expectationoperator. Thus, Mγ(s) =

∫∞0e−sxfγ(x)dx can be derived

as

Mγ(s) =N∑i=1

αiΓ(βi)

(s+ ζi)βi. (3)

D. Moments

The rth moment associated with the MG distribution,mγ(r), can be calculated as mγ(r) = E(γr), to yield

mγ(r) =N∑i=1

αiΓ(βi + r)ζ−(βi+r)i . (4)

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The mathematically tractable expressions (1)-(4) demon-strate the major benefit of the MG distribution. The perfor-mance metrics have convenient expressions, i.e., no compli-cated special functions are required. This fact can facilitate theperformance studies enormously. If a given wireless channelcan be represented as an MG distribution, the common per-formance metrics such as error rates, outage and others areimmediately derived, with details given in Section V.

III. MG DISTRIBUTION FOR TYPICAL WIRELESSCHANNELS

This section shows how to represent the SNR PDF of theNL, K, KG, κ-µ, Nakagami-q (Hoyt), η-µ and Nakagami-n(Rician) channel models in the form of an MG model, as in(1).

A. NL Composite Channel

The SNR distribution of the NL channel is a gamma-lognormal (GL) distribution, given as [26]

fγ(x) =

∫ ∞

0

xm−1e−mxρy

Γ(m)

(m

ρy

)me−

(ln y−µ)2

2λ2

√2πλy

dy (5)

where m is the fading parameter in Nakagami-m fading, ρ isthe unfaded SNR, and µ and λ are the mean and the standarddeviation of the lognormal distribution, respectively. Whenm = 1, expression (5) is the SNR distribution of the RLdistribution. The fading and shadowing effects diminish forlarger m and smaller λ, respectively. A closed-form expressionof the composite GL SNR distribution is not available in theliterature.

By using substitution t = ln y−µ√2λ

, expression (5) can bewritten as

fγ(x) =xm−1

√π Γ(m)

(m

ρ

)m ∫ ∞

−∞e−t2g(t)dt (6)

where g(t) = e−m(√2λt+µ)e−

mρ e−(

√2λt+µ)x. The integration in

(6), I =∫∞−∞ e−t2g(t)dt, is a Gaussian-Hermite integration

which can be approximated as I ≈∑N

i=1 wig(ti) where tiand wi are the abscissas and weight factors for the Gaussian-Hermite integration [27]. Therefore, we can express fγ(x) in(6) as the MG distribution given in (1). After normalizationof∫∞0fγ(x)dx = 1, we find

αi =ψ(θi, βi, ζi), βi = m, ζi =m

ρe−(

√2λti+µ),

θi =

(m

ρ

)mwie

−m(√2λti+µ)

√πΓ(m)

(7)

where ψ(θi, βi, ζi) = θi∑Ni=1 θiΓ(βi)ζ

−βii

. Function ψ(θi, βi, ζi)

is also used for subsequent cases.

B. K and KG Channels

The SNR distribution of the KG channel has a closed-formexpression with the nth-order modified Bessel function ofthe second kind [9]. With some mathematical simplifications,

the SNR distribution of the KG channel, which is a gamma-gamma distribution, can be rewritten in an integral form as

fγ(x) =λmxm−1

Γ(m)Γ(k)

∫ ∞

0

e−tg(t)dt (8)

where g(t) = tα−1e−λxt , λ = km

γ and α = k − m. Here kand m are the distribution shaping parameters, which representthe multipath fading and shadowing effects of the wirelesschannel, respectively. The integral in (8), I =

∫∞0e−tg(t)dt,

can be approximated as a Gaussian-Laguerre quadrature sumas I ≈

∑Ni=1 wig(ti) where ti and wi are the abscissas

and weight factors for the Gaussian- Laguerre integration[27]. Thus, (8) can be written as the MG distribution withparameters

αi = ψ(θi, βi, ζi), βi = m, ζi =λ

ti, θi =

λmwitα−1i

Γ(m)Γ(k). (9)

C. η-µ Channel

The η-µ channel model is a generalized form to model thenon-line of sight small-scale fading of a wireless channel [28].The Rayleigh, Nakagami-m and Hoyt distributions are specialcases of the η-µ channel model. The η-µ SNR distribution isgiven as [29]

fγ(x) =2√πµµ+ 1

2hµxµ−12 e

−2µhxγ

Γ(µ)Hµ− 12 γµ+

12

Iµ− 12

(2µHx

γ

)(10)

where the parameter µ =(1+H2/h2)E2(γ)

2V ar(γ) (µ > 0) representsthe number of multipath clusters (V ar(·) represents the vari-ance), and Iv(·) is the vth-order modified Bessel function ofthe first kind. Parameters h and H are to be explained inthe following. The η-µ channel includes two fading formats,Format 1 and Format 2, for two different physical represen-tations. In Format 1, the independent in-phase and quadraturecomponents of the fading signal have different powers, and η(0 < η < ∞) is the power ratio of the in-phase componentto the quadrature component. Two parameters h and H aredefined as h = 2+η−1+η

4 and H = η−1−η4 , respectively. In

Format 2, the in-phase and quadrature components of thefading signal are correlated and have identical powers. η(−1 < η < 1) is the correlation coefficient between the in-phase and quadrature components. Two parameters h and Hare defined as h = 1

1−η2 and H = η1−η2 , respectively [28].

Only a few performance studies for the η-µ channel havebeen published in the literature, probably because the modifiedBessel function of the first kind in (10) leads to mathematicalcomplexity [29]–[33]. In the following, the η-µ SNR distribu-tion is approximated by using the MG distribution.

For a real number v, the function Iv(z) can be computedusing [25]

Iv(z) =∞∑k=0

1

k!Γ(v + k + 1)

(z2

)2k+v

. (11)

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Therefore, the η-µ SNR distribution (10) can be given in analternative form as

fγ(x) =2√πµµ+ 1

2hµe−2µhx

γ

Γ(µ)Hµ− 12 γµ+

12

∞∑i=1

(µHγ

)2i+µ− 52

x2µ−3+2i

(i− 1)!Γ(µ+ i− 12 )

.

(12)The required accuracy1 to approximate the exact fγ(x) can beachieved by summing a finite number, N , of terms in (12). Bymatching the two PDFs given in (10) and (12), the parametersof the MG distribution can be evaluated as

αi =ψ(θi, βi, ζi), βi = 2(µ− 1 + i), ζi =2µh

γ,

θi =2√πµµ+ 1

2hµ

Γ(µ)Hµ− 12 γµ+

12

(µHγ

)2i+µ− 52

(i− 1)!Γ(µ+ i− 12 ).

(13)

Alternatively, the vth-order modified Bessel function of thefirst kind, Iv(z), can be approximated by using the integralrepresentation [25, eq. (8.431.5)]

Iv(z) =

∫ π

0

ez cosϑ cos(vϑ)dϑ

π−∫ ∞

0

sin(vπ)e−z cosh t−vtdt

π.

(14)With ϑ = uπ

2 + π2 and vt = p, Iv(z) can be

further written as Iv(z) = I1 − I2, where I1 =∫ 1

−1g1(u)du is a Gaussian-Legendre integration, and I2 =∫∞

0e−pg2(p)dp is a Gaussian-Laguerre integration where

g1(u) = 12e

−z sin(πu2 ) cos

((u+ 1)πv2

)and g2(p) =

sin(πv)e−z cosh( pv )/(πv). Similar to Section III-A with

Gaussian-Hermite integration, the SNR distribution of the η-µchannel can be approximated by the MG model.

D. Nakagami-q (Hoyt) Channel

Satellite links with strong ionospheric scintillation can bemodeled with the Nakagami-q distribution, and the SNRdistribution of the Nakagami-q channel is given as [26]

fγ(x) =1 + q2

2qγe− (1+q2)2

4q2γxI0

(1− q4

4q2γx

)(15)

where I0(·) is the zeroth-order modified Bessel function ofthe first kind. The fading parameter q varies from 0 to 1,where q = 0 and q = 1 represent the one-sided Gaussian andRayleigh distributions, respectively. Further, this distribution isa special case of the η-µ distribution when µ = 1

2 and η = q2.Using Format 1 of the η-µ distribution, the parameters of theMG distribution for the Nakagami-q channel can be derivedfrom (13) to yield

αi =ψ(θi, βi, ζi), βi = 2i− 1, ζi =(1 + q2)2

4q2γ,

θi =(1 + q2)

2qγΓ(i)(i− 1)!

(1− q4

8q2γ

)2i−2

.

(16)

1The required accuracy can be defined in terms of the mean-square error(MSE) between the exact and approximated expressions or by matching thefirst r moments.

E. κ-µ Channel

The κ-µ distribution fits well with channels having line-of-sight components. Nakagami-n (Rician) and Nakagami-mchannels are special cases of the κ-µ channel. The κ-µ SNRdistribution is [29]

fγ(x) =µ(1 + κ)

µ+12

κµ−12 eµκγ

µ+12

· xµ−12 e−

µ(1+κ)γ xIµ−1

(2µ

√κ(1 + κ)

γx

) (17)

where κ (κ > 0) is the power ratio of the dominant compo-nents to the scattered components of the signal, and µ (µ > 0)is defined as µ = (1+2κ)E2(γ)

(1+κ)2V ar(γ) . Since fγ(x) includes themodified Bessel function of the first kind with the square rootof the random parameter x, it is difficult to obtain the MG formwith one of the Gaussian integration methods, as discussed inprevious subsections. To address this difficulty, the κ-µ SNRdistribution given in (17) can be written using (11) as

fγ(x) =µ(1 + κ)

µ+12

κµ−12 eµκγ

µ+12

∞∑i=1

[µ2i+µ−3

Γ(µ− 1 + i)(i− 1)!

·(κ(1 + κ)

γ

) 2i+µ−32

xµ+i−2e−µ(1+κ)

γ x

].

(18)

The required accuracy for approximating the exact fγ(x) canbe achieved by summing a finite number, N , of terms in(18). By matching the two PDFs given in (17) and (18), theparameters of the MG distribution can be evaluated as

αi =ψ(θi, βi, ζi), βi = µ+ i− 1, ζi =µ(1 + κ)

γ,

θi =µ(1 + κ)

µ+12

κµ−12 eµκγ

µ+12

µ2i+µ−3(

κ(1+κ)γ

) 2i+µ−32

Γ(µ− 1 + i)(i− 1)!.

(19)

Alternatively, one can use a different approach in which theMGF of SNR under κ − µ distribution can be matched withthe MGF of SNR under the MG distribution given in (3).Using the power series expansion of the exponential functionex =

∑∞n=0

xn

n! , the MGF of the κ-µ SNR distribution givenin [30] can be re-written as an infinite form. By matching thetwo MGFs, the parameters of (1) can be evaluated. Details areomitted due to space limit.

F. Nakagami-n (Rician) Channel

The Nakagami-n or Rician channel model fits well withchannels having a strong line-of-sight component. The corre-sponding SNR distribution is given as [26]

fγ(x) =(1 + n2)e−n2

γe−

(1+n2)γ xI0

(2n

√(1 + n2)

γx

)(20)

where n is the fading parameter (0 ≤ n <∞), and the Ricianfactor K is given as K = n2. The Nakagami-n distribution isa special case of the κ-µ distribution when µ = 1 and κ = n2.

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1 2 3 4 5 6 7 8 9 10 11

10−4

10−3

10−2

10−1

N

MS

E

KG

GGammaMG

Fig. 1: The MSE versus N when the GL distribution isapproximated by KG, G, Gamma and MG for m = 2.7, λ = 1,µ = 0, and the average SNR = 0 dB.

Therefore, the parameters of the MG distribution given in (1)can be evaluated as

αi =ψ(θi, βi, ζi), βi = i, ζi =(1 + n2)

γ,

θi =(1 + n2)

en2 [(i− 1)!]2γ

(n2(1 + n2)

γ

)i−1

.

(21)

G. Rayleigh and Nakagami-m Channels

The SNR distributions of the Rayleigh and Nakagami-mchannels are exponential and gamma distributions, respectively[26, eq. (2.7) and (2.21)]. The two distributions are specialcases of the MG distribution. When the Rayleigh distributionis written in the MG form given in (1), the correspondingparameters are N = 1, α1 = 1

γ , β1 = 1 and ζ1 = 1γ . For the

Nakagami-m distribution, the corresponding parameters areN = 1, α1 = mm

Γ(m)γm , β1 = m and ζ1 = mγ .

IV. DETERMINATION OF THE NUMBER N IN THE MGDISTRIBUTION

For the MG distribution to approximate other channelmodels, the number of components N needs to be determined.This can be selected as the minimum value such that (i) themean-square error (MSE) or KullbackLeibler (KL) divergencebetween the target distribution and the MG distribution isbelow a threshold; or (ii) the first r moments of the twodistributions match.

A. Accuracy of MG Distribution to Approximate WirelessChannel SNR

In the literature, the composite GL model has been ap-proximated by KG, G and Gamma models. Here, we com-pare the accuracy of the MG approximation with that of

1 2 3 4 5 6 7 8 9 10 1110

−4

10−3

10−2

10−1

100

N

KL

dive

rgen

ce

K

G

GGammaMG

Fig. 2: The KL divergence (DKL) versus N when the GLdistribution is approximated by KG, G, Gamma and MG form = 2.7, λ = 1, µ = 0, and the average SNR = 0 dB.

those approximations. One of the possible measures of ac-curacy is the MSE between two PDFs: the approximatePDF fApp(x) and the exact PDF fExt(x). It is definedas MSE = E

[(fExt(x)− fApp(x))

2]. Another possible

measure of accuracy is the KL divergence (DKL) be-tween fApp(x) and fExt(x), which is defined as DKL =∫∞−∞ fExt(x) log

fExt(x)fApp(x)

dx.2

We select KG, G and Gamma distributions for the compari-son. The Gamma approximation is obtained by approximatingKG PDF by a Gamma PDF [8]. The MSEs and DKLs betweenthe GL and its MG approximation (eq. (7)), GL and KG, GLand G, and GL and Gamma can be calculated numerically, asshown in Fig. 1 and Fig. 2, respectively, for m = 2.7, λ = 1,µ = 0, and the average SNR = 0 dB. The parameters of KG, Gand Gamma distributions to match the target GL distributionare obtained from [5], [7], [8]. The MSE and DKL betweenGL and MG distributions are less than 10−3 when the numberof components N ≥ 6 and N ≥ 8, respectively. Based onMSE, MG model is better than Gamma, KG and G modelswhen N ≥ 2, 4 and 5, respectively. Based on KL divergence,MG model is better than Gamma, KG and G models whenN ≥ 3, 4 and 6, respectively. It can be seen that MSE and KLdivergence give similar results for the minimum value of Nthat makes the MG model more accurate than Gamma, KG

or G model. Further, these MSE and DKL with MG modeldecrease significantly as N slightly increases.

This fact is also evidenced by Fig. 3, which shows theCDFs of the GL and its approximations KG, G, Gamma and

2Although both the mean-square error (MSE) and the Kullback-Leibler(KL) divergence are measures of the difference between two PDFs, theygive different measurements. Nevertheless, they give similar results for theminimum value of N that makes the MG model more accurate than Gamma,KG or G model, as shown subsequently.

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10−4

10−3

10−2

10−1

100

101

1e−6

1e−5

1e−4

0.001

0.01

0.10.20.30.40.50.60.70.80.9

0.99

0.999

1−(1e−4)

x

CD

F

GLK

GGGammaMG

Fig. 3: The exact CDF of GL distribution and the CDFs ofthe KG, G, Gamma and MG approximations. The parametervalues are m = 2.7, λ = 1, µ = 0, and the average SNR = 0dB. The number of components in the MG model is N = 5.

MG. These curves are plotted on a GL paper. The ordinateof the GL paper is obtained using the transformation F−1

GL (t)where FGL(x) is the CDF of GL distribution. Thus the GLdistribution is a straight line on the GL paper, and othersare not. The inverse function is numerically calculated usingMATLAB. The following observations are made:

1) The exact GL CDF (solid line) matches perfectly withthe MG approximation (small circles) for all x. Just N =5 terms have been used in this case. Even better accuracyis possible by slightly increasing N .

2) The KG approximation [6] deviates significantly in thelower tail (x < 0.09) and also in the upper tail (x > 5).

3) The G approximation [7] deviates in the lower tail (x <0.1).

4) The Gamma approximation [8] deviates significantly inboth lower tail (x < 1) and the upper tail (x > 5).

Clearly, the MG distribution is a more accurate representationof composite fading channels.

Similarly, Fig. 4 shows the SNR distributions of the KG,Nakagami-q (Hoyt), η-µ (Format 1), Rician and κ-µ channelmodels and their corresponding approximations in the MGform, when the value of N in the MG distribution is selectedas the minimum value that satisfies MSE ≤ 10−6. Excellentmatch is also observed in all curves. Note that in Fig. 4 andsubsequent figures, the continuous lines and discrete markersshow the curves corresponding to the exact distribution andthe approximated MG distribution, respectively.

B. Moment Matching

The parameters of αi, βi and ζi in the MG distributioncan be determined based on matching the MGFs of the exactdistribution and the MG distribution. For brevity, we determine

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x

f γ(x)

K

G (m = 2, k = 3, N = 15)

Hoyt (q2 = 0.5, N = 3)η−µ (η = 3.5, µ = 10, N = 8)

Rician (n2=3, N = 16)κ−µ (κ = 3, µ = 2, N = 16)

Fig. 4: Exact SNR distributions of KG, Nakagami-q (Hoyt),η-µ (Format 1), Rician and κ-µ channel models and their MGapproximations.

(κ, µ)

(3, 0.5) (3, 1) (3, 2) (7, 0.5) (7, 1) (7, 2)

1st moment 3 3 3 3 3 3

2nd moment 19 14 12 15 12 11

3rd moment 154 83 55 87 57 43

N 8 11 16 12 20 26

TABLE I: The selected value of N and the nearest integer val-ues of the first three moments of both exact and approximatedSNR distributions of κ-µ channel.

the value of N as the minimum value such that the first rmoments of the two distributions have the same nearest integervalues. Table I shows the selected N value when r = 3 forκ-µ distribution. The exact distribution and the approximatedMG distribution have the same nearest integer values for thefirst 3 moments, which are also shown in Table I. In Table I,the (3, 1) and (7, 1) columns are corresponding to Nakagami-n distribution (Rician) with Rician factors K = 3 and K = 7,respectively.

To determine the parameters of the MG distribution toapproximate other channel models, the exact SNR momentexpressions (mγ(r) = E(γr)) for the Nakagami-lognormal,KG, Nakagami-q and Nakagami-n are available in the litera-ture [9], [26], [34], and the exact SNR moment expressions ofη-µ and κ-µ distributions can be derived from their momentsof the envelop distribution given in [28, eqs. (5), (43), and(46)].

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C. Complexity of Determining N

The number of terms N in the MG model may be deter-mined iteratively. N can be increased until the MSE or KL di-vergence requirements are met. In our numerical result shownin Figs. 1 and 2, approximately N = 8 can meet the accuracyrequirement of 10−3. To achieve 10−6 accuracy (although wemay not need that level of accuracy), approximately N = 15is needed.

Each iteration (corresponding to a particular N ) requires3N parameters αi, βi and ζi (i = 1, · · · , N ). If Gaussianintegration methods (Gaussian-Legendre, Gaussian- Laguerre,or Gaussian-Hermite) are used, their abscissas and weightfactors are already tabulated (e.g., in [27]), or can be generatedefficiently by using simple MATLAB codes. Note that specialfunctions are not involved in the calculations of the parameters.

V. PERFORMANCE ANALYSIS BASED ON THE MGCHANNEL MODEL

Performance analysis of wireless technologies such asMIMO, cooperative communications, cognitive radio andultra-wideband (UWB) radio has become important recently.The MG distribution helps to provide a unified performanceanalysis framework, because of the mathematical tractabilityof its CDF, MGF and moments (Section II) and because of itsversatility (Section III). To this end, this section shows how theMG distribution allows the derivation of typical performancemetrics such as error rate, outage, and others.

A. Diversity Order and Array Gain

The diversity order is the magnitude of the slope of theerror probability versus SNR curve (log-log scale) in the highSNR region. The array gain measures the shift of the errorprobability curve to the left. The diversity order and the arraygain relate to the asymptotic value of the MGF near theinfinity, i.e., if the MGF, Mγ(t), can be written in the form

|Mγ(t)| = b|t|−d +O(|t|−(d+1)) as t→ ∞

then b and d define the array gain and diversity order, respec-tively [35]. Clearly, using the binomial series expansion, (3)can be rewritten as

|Mγ(s)| =N∑i=1

αiΓ(βi)

(s−βi +

∞∑k=1

(−βik

)ζki s

−(βi+k)

).

(22)Therefore, the array gain is b ≈ αnΓ(βn) and the diversityorder is d = βn, where n is the index of the first nonzero αi,i.e., αi = 0 ∀i < n, and αn = 0. Accordingly, the diversityorders of NL, K, KG, η-µ, Hoyt, κ-µ, Rician, Rayleigh andNakagami-m fading channels are m, 1, m, 2µ, 1, µ, 1, 1, andm , respectively.

B. Average Channel Capacity

By using Shannon’s theorem, the average channel capacityof a single-input single-output (SISO) channel, C, can becalculated by averaging the instantaneous channel capacityover the SNR distribution as C =

∫∞0B log2(1+x)fγ(x)dx,

where B is the signal transmission bandwidth. If βi is aninteger, the average channel capacity over the MG distribution,C, can be calculated by using results in [36], as

C =B

ln 2

N∑i=1

αi(βi − 1)!eζiβi∑k=1

Γ(k − βi, ζi)

ζki(23)

where Γ(·, ·) is the upper incomplete gamma function definedas Γ(a, ρ) ,

∫∞ρta−1e−tdt [25, eq. (8.350.2)]. Next we

provide a method to calculate C for any value of βi. Byreplacing log2(1 + x) with the Meijer’s G-function [37, eq.(01.04.26.0003.01)], C can be evaluated in closed-form, whichis valid for any βi, as

C =B

ln 2

N∑i=1

αiζ−βi

i G1,33,2

[ζ−1i

∣∣∣∣ 1− βi, 1, 11, 0

]. (24)

For integer βi, both expressions in (23) and (24) are equalnumerically.

C. Average Symbol Error Rate (SER)

Since we have a MGF without special functions in the MGchannel model, it can be used to evaluate the average SER ofM -PSK, M -QAM and M -AM, as follows.

1) M-PSK: The average SER for M -PSK, P pske , is given in

[26, eq. (9.15)] for some channel models. With the MGF givenin (3), the average SER for M -PSK over the MG distributionP pske can be evaluated as

P pske =

N∑i=1

αiΓ(βi)

πζiβi

∫ (M−1)πM

0

(sin2 θ

sin2 θ +gpskζi

)βi

dθ (25)

where gpsk = sin2( πM ). Therefore, the average SER of the

M -PSK modulation can be evaluated in closed-form for anyvalue of βi with the aid of [38, eq. (10)].

2) M-QAM: Square M -QAM signals with a constellationsize M = 2k with even k values are considered. The averageSER for M -QAM, P qam

e , is given in [26, eq. (9.21)] for somechannel models. When the MG distribution is used, P qam

e canbe evaluated as

P qame =

N∑i=1

KαiΓ(βi)

ζiβi

[∫ π2

0

(sin2 θ

sin2 θ +gqam

ζi

)βi

dθ

−√M − 1√M

∫ π4

0

(sin2 θ

sin2 θ +gqam

ζi

)βi

dθ

] (26)

where gqam = 32(M−1) and K = 4

π (1−1√M) . P qam

e in (26)can be evaluated in closed-form for any value of βi with theaid of [38, eq. (12)].

3) M-AM: Similarly, the average SER for M -AM, P ame , is

given in [26, eq. (9.19)] for some channels. If it is evaluatedbased on the MG distribution, we have

P ame =

2(M − 1)

πM

N∑i=1

αiΓ(βi)

πζiβi

∫ π2

0

(sin2 θ

sin2 θ + gam

ζi

)βi

dθ

(27)where gam = 3

(M2−1) . With the aid of [26, eq. (5A.1)] or [38,eq. (5)], P am

e can be evaluated in closed-form for any valueof βi.

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS (ACCEPTED) 8

Similarly, the SER analysis for other modulation schemesover different digital communication systems, for example,as given in [39], [40], can be performed using the MGdistribution.

D. Outage Probability

The outage probability, which is the probability that thereceived SNR is below a given threshold γth, can easily becalculated as Pout = Fγ(γth), where Fγ(x) is given in (2).

E. Energy Detection in Cognitive Radio

In cognitive radio networks, energy detection of a primarysignal is essential. The detection performance is typicallyillustrated by using the receiver operating characteristic (ROC)curve. Although such performance analysis for the Rayleigh,Nakagami-m, Rician and η-µ fading channels is available in[41], [42], the results for the Rician channel are limited. Thisis due to the detection probability being expressed by thegeneralized Marcum-Q function with limited analytical results.This problem can be circumvented by using the MG model,as we will illustrate next.

1) Average Detection Probability: When a primary signalexists, the detection probability is defined as the probabil-ity that the received energy is higher than a pre-definedthreshold λ. The probability of detection (Pd) is expressedas Pd = Qu(

√2γ,

√λ), where Qu(·, ·) is the generalized

Marcum-Q function. The number of samples, u, is an in-teger value [41]. The average detection probability, Pd =∫∞0Qu(

√2x,

√λ)fγ(x)dx, can further be re-written by re-

placing the Marcum-Q function by its circular contour integralrepresentation [31] to yield

Pd =e−

λ2

j2π

∮Γ

Mγ

(1− 1

z

)e

λ2 z

zu(1− z)dz (28)

where Γ is a circular contour with radius r ∈ [0, 1). Aftersubstituting the MGF given in (3), the average detectionprobability can be re-written as

Pd = e−λ2

N∑i=1

αiΓ(βi)

(ζi + 1)βi

1

j2π

∮Γ

g(z)dz (29)

where

g(z) =e

λ2 z

zu−βi(1− z)(z − 1

1+ζi

)βi.

We can solve the contour integral by applying the ResidueTheorem assuming integer values for βi. There are two pos-sible scenarios, u > βi and u ≤ βi.

When u > βi: There are (u − βi) poles at z = 0 and βipoles at z = 1

1+ζi. Therefore, Pd can be calculated as

Pd = e−λ2

N∑i=1

αiΓ(βi)

(ζi + 1)βi

[Res (g; 0) + Res

(g;

1

1 + ζi

)](30)

where Res (g; 0) and Res(g; 1

1+ζi

)are the residues of g(z)

at z = 0 and z = 11+ζi

, respectively. Further, Res (g; 0) and

Res(g; 1

1+ζi

)can be evaluated as

Res (g; 0) =

[du−βi−1

dzu−βi−1 g(z)zu−βi

] ∣∣∣∣z=0

(u− βi − 1)!

Res(g;

1

1 + ζi

)=

[dβi−1

dzβi−1 g(z)(z − 11+ζi

)βi

] ∣∣∣∣z= 1

1+ζi

(βi − 1)!.

When u ≤ βi: There are βi poles at z = 11+ζi

. Therefore,Pd can be calculated as

Pd = e−λ2

N∑i=1

αiΓ(βi)

(ζi + 1)βi

Res(g;

1

1 + ζi

). (31)

2) Area Under the ROC Curve (AUC): The area underthe ROC curve (AUC) is another method to describe theoverall energy detector performance, which has been recentlyintroduced to the wireless communication field [43]. The AUCof an energy detector for a specific value of instantaneous SNRγ, A(γ), is derived as [43]

A(γ) =1−u−1∑k=0

1

2k k!γke−

γ2

+u−1∑

k=1−u

Γ(u+ k)

2u+kΓ(u)e−γ

1F1

(u+ k; 1 + k;

γ

2

) (32)

where 1F1(·; ·; ·) is the regularized confluent hypergeometricfunction of 1F1(·; ·; ·) [37]. The average AUC under the MGdistribution, A, can be derived as A =

∫∞0A(x)fγ(x)dx to

yield

A =1−u−1∑k=0

1

2kk!

N∑i=1

αiΓ(k + βi)

( 12 + ζi)k+βi+

u−1∑k=1−u

Γ(u+ k)

2u+kΓ(u)

N∑i=1

αiΓ(βi)

(1 + ζi)βi2F1

(βi;u+ k; 1 + k;

1

2(1 + ζi)

)(33)

where 2F1(·; ·; ·; ·) is the regularized confluent hypergeometricfunction of 2F1(·; ·; ·) [37]. This derivation is very similar tothe derivation given in [43] for Nakagami-m fading.

Note that the performance of cooperative relay channelshas been extensively studied over Rayleigh and Nakagami-mfading channels (e.g., [44], [45]). However, analysis may belacking for lognormal shadowing, Rician, Hoyt, K, η-µ and κ-µ fading channels. The MG distribution may help in this case.For example, in [46], energy detector performance is analyzedfor cooperative spectrum sensing in a cognitive radio networkover fading and shadowing, which are modeled by using theMG distribution. The SER analysis of an amplify-and-forwardrelay network [45] and optimal resource allocation [47] aretwo potential applications of the MG model. Due to the spacelimitation, these topics are omitted here.

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−10 −5 0 5 10 15 200

1

2

3

4

5

6

7

Average SNR (dB)

Nor

mal

ized

ave

rage

cha

nnel

cap

acity

(b/

s/H

z)

NL, (m,λ,µ)=(2,1,0.25) K

G, (m,k)=(2,5)

η−µ, (η,µ)=(0.5,1.5)

Hoyt, q2=0.5κ−µ, (κ,µ)=(7,2)

Rician, n2=3

Fig. 5: The average capacity of an SISO channel versusaverage SNR over different fading channels.

0 10 20 30 40 5010

−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Average SNR (dB)

Out

age

prob

abilt

y

NL, (m,λ,µ)=(2,1,0.25)K

G, (m,k)=(2,5)

η−µ, (η,µ)=(0.5,1.5)

Hoyt, q2=0.5κ−µ, (κ,µ)=(7,2)

Rician, n2=3

Fig. 6: The outage probability of an SISO channel versusaverage SNR over different fading channels for γth = 0 dB.

F. Numerical Results

Two main focuses of this sub-section are (1) to showhow the performance analysis based on the MG approx-imation matches with the exact results, and (2) to com-pare the performance of different fading channels. Wechoose typical distribution parameters. The value of Nis selected as the minimum value to satisfy MSE ≤10−6. As an example, the parameters and N for the NL,KG, η-µ (Format 1), Hoyt, κ-µ and Rician channels arechosen as (m,λ, µ,N )=(2, 1, 0.25, 10), (m, k,N )=(2, 5, 6),(η, µ,N )=(0.5, 1.5, 5), (q2, N )=(0.5, 3), (κ, µ,N )= (7, 2, 36)and (n2, N )=(3, 16), respectively.

The performance curves for the average channel capacity,

−10 0 10 20 30 40

10−10

10−8

10−6

10−4

10−2

Average transmit SNR (dB)

SE

R

NL, (m,λ,µ)=(2,1,0.25) K

G, (m,k)=(2,5)

η−µ, (η,µ)=(0.5,1.5)

Hoyt, q2=0.5κ−µ, (κ,µ)=(7,2)

Rician, n2=3

BPSK

QAM

Fig. 7: The SER of an SISO channel versus average SNR overdifferent fading channels for BPSK and QAM.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pf

Pd

NL, (m,λ,µ)=(2,1,0.25) K

G, (m,k)=(2,5)

η−µ, (η,µ)=(0.5,1.5)

Hoyt, q2=0.5κ−µ, (κ,µ)=(7,2)

Rician, n2=3

0 dB

5 dB

Fig. 8: The receiver operating characteristic (ROC) curves ofan energy detector over different fading channels for averageSNR γ = 0 dB, 5 dB and u = 1.

the outage probability, average SER for BPSK and QAM, theROC and the complementary AUC (CAUC = 1-AUC) of anenergy detector are plotted in Figs. 5-9, respectively, basedon both exact (continuous lines) and approximated (discretepoints) MG distributions. All figures show an excellent match.

As discussed in Section V-A, the achievable diversity or-ders of NL, K, KG, η-µ, Hoyt, κ-µ, Rician, Rayleigh andNakagami-m fading channels are m, 1, m, 2µ, 1, µ, 1, 1, andm, respectively. The diversity order can be illustrated by usingoutage probability (Fig. 6), SER (Fig. 7) or complementaryAUC (Fig. 9) versus average SNR plots in the high SNRregion. From the figures, NL and KG models show diversity

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0 10 20 30 40 5010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Average SNR (dB)

CA

UC

NL, (m,λ,µ)=(2,1,0.25) K

G, (m,k)=(2,5)

η−µ, (η,µ)=(0.5,1.5)

Hoyt, q2=0.5κ−µ, (κ,µ)=(7,2)

Rician, n2=3

Fig. 9: The average complementary area under the ROC curves(CAUC = 1 − AUC) of an energy detector versus averageSNR over different fading channels for u = 1.

order 2 because their fading parameters are m = 2. Hoyt andRician models always have diversity order 1. Since η-µ andκ-µ models have µ = 1.5 and µ = 2, they have diversityorder 3 and 2, respectively. All these confirm accuracy of ouranalysis in Section V-A. Further, we compare the performanceof η-µ and Hoyt channels. Although both channels have samepower ratios in our numerical examples (i.e., η=q2=0.5), the η-µ channel has higher effective multipath clusters, which help toachieve a diversity order of 3 (=2µ) while the Hoyt channel hasdiversity order of one. Therefore, performance of η-µ channelwith (η, µ)=(0.5,1.5) is better than the performance of Hoytchannel with q2=0.5 in terms of channel capacity, outage,SER, and energy detection capability (Figs. 5-9). Similarly,we can compare performance of κ-µ and Rician channels.For the two channels in our numerical examples, the powerratios of the dominant components to the scattered componentsof the signal are κ = 7 and n2 = 3, respectively, and thediversity orders are 2 and 1, respectively. So performance ofκ-µ channel with (κ, µ)=(7,2) is better than the performanceof Rician channel with n2=3 in terms of channel capacity,outage, SER, and energy detection capability (Figs. 5-9).Since NL, KG, η-µ, and κ-µ channel models do not havestraightforward relationships among each other, no clear-cutperformance comparison can be done.

VI. CONCLUSIONS

The MG distribution to model the SNR of the wirelesschannels has been proposed. Theoretical results [23] [48] showit converges to any PDF over (0,∞), a justification of thismodel. It is not only ideal for composite channels, but alsoeffective for small-scale fading channels. The parameters ofthe MG distribution to match a target distribution can beobtained by approximating with Gaussian quadrature formulas,by matching moments (MGF) or by matching PDFs. We

demonstrate that the MG model offers a more accurate repre-sentation of composite fading channels than those provided bythe K models and other alternatives, which have recently beenused in wireless research. Due to its mathematically tractableform and high accuracy, the MG distribution thus allows rapidevaluation of performance metrics such as channel capacity,outage, error rate and others of MIMO systems, cooperativerelay channels, cognitive radio, UWB and others. Furtherresearch directions include the performance analysis of otherwireless systems and model fitting based on measured channeldata. We believe that the MG distribution opens up these andother research opportunities.

ACKNOWLEDGMENTS

The authors would like to thank the Editor Dr. Man-On Punand the anonymous reviewers for their constructive commentswhich improve the presentation of this paper.

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Saman Atapattu (S’06) received the B.Sc. degreein electrical and electronics engineering from theUniversity of Peradeniya, Sri Lanka in 2003 and theM. Eng. degree in telecommunications from AsianInstitute of Technology (AIT), Thailand in 2007. Heis currently working towards the Ph.D. degree inelectrical and computer engineering at the Universityof Alberta, Edmonton, AB, Canada.

His research interests include cooperative commu-nications, cognitive radio networks, and performanceanalysis of communication systems. He has been

awarded the Izaak Walton Killam Memorial Scholarship (2011-2013) and theAlberta Innovates Graduate Student Scholarship (2011-2012).

Chintha Tellambura (F’11) received the B.Sc. de-gree (with first-class honor) from the University ofMoratuwa, Sri Lanka, in 1986, the M.Sc. degree inElectronics from the University of London, U.K., in1988, and the Ph.D. degree in Electrical Engineeringfrom the University of Victoria, Canada, in 1993.

He was a Postdoctoral Research Fellow with theUniversity of Victoria (1993-1994) and the Univer-sity of Bradford (1995-1996). He was with MonashUniversity, Australia, from 1997 to 2002. Presently,he is a Professor with the Department of Electrical

and Computer Engineering, University of Alberta. His research interests focuson communication theory dealing with the wireless physical layer.

Prof. Tellambura is an Associate Editor for the IEEE TRANSACTIONSON COMMUNICATIONS and the Area Editor for Wireless CommunicationsSystems and Theory in the IEEE TRANSACTIONS ON WIRELESS COMMU-NICATIONS. He was Chair of the Communication Theory Symposium inGlobecom’05 held in St. Louis, MO.

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS (ACCEPTED) 12

Hai Jiang (M’07) received the B.Sc. and M.Sc.degrees in electronics engineering from Peking Uni-versity, Beijing, China, in 1995 and 1998, respec-tively, and the Ph.D. degree (with an OutstandingAchievement in Graduate Studies Award) in elec-trical engineering from the University of Waterloo,Waterloo, Ontario, Canada, in 2006.

Since July 2007, he has been a faculty memberat the University of Alberta, Edmonton, Alberta,Canada, where he is currently an Associate Professorat the Department of Electrical and Computer Engi-

neering. His research interests include radio resource management, cognitiveradio networking, and cross-layer design for wireless multimedia communi-cations.

Dr. Jiang is an Associate Editor for the IEEE TRANSACTIONS ON VE-HICULAR TECHNOLOGY. He served as a Co-Chair for the Wireless andMobile Networking Symposium at the IEEE International Conference onCommunications (ICC) in 2010. He received an Alberta Ingenuity NewFaculty Award in 2008 and a Best Paper Award from the IEEE GlobalCommunications Conference (GLOBECOM) in 2008.